Solving Employee Timetabling Problems by Generalized Local Search

1
Solving Employee Timetabling Problems by
Generalized Local Search

• Amnon Meisels
• University of Udine, Italy
• and
• Ben-Gurion University

2
Employee Timetabling Problems

• Call centers, such as CellCom, employ different
  types of operators to work in many different
  shifts per day and week
• Shifts have a variety of start and end times and
  consist of different tasks
• Operators have to be assigned to shifts over the
  week so that all the required tasks are assigned
• Typical operators are part-time students that
  have a weekly schedule of their studies
• A weekly timetable - meets requirements and does
  not violate personal constraints

3
ETPs - Constraints

• Requirements Shifts need a required number of
  assigned employees to each of their tasks
• -- Two Senior_Nurses in Morning_Shifts
• Ability Employees are assigned to tasks,
  according to their abilities
• -- Certified nurses can be assigned to
  Head_Nurse
• Availability Personal preferences of employees
  restrict their assignment to only a subset of
  shifts
• -- Senior doctors are not assigned to
  Friday_Shifts

4
ETPs - Constraints (II)

• Conflicts Employees cannot be assigned to two
  conflicting shifts
• -- No Morning_Shift following a Night_Shift
• Workload There are bounds on the number of
  tasks assigned to each employee
• -- Maximum 10 hours a day, 40 hours per week
• Regulations Certain limits are imposed on the
  number of specific tasks assigned to employee
• -- at most 3 Night_Shifts in 2 weeks

5
ETPs as Constraint Networks

• Variables are shift-task pairs ltSj ,Tkgt
• Employees are the assigned values Ei
• Unavailabilities remove values from domains
• Conflicts are binary constraints
• Limits on number of assignments, either general
  or specific, are cummulative constraints
• Typical real-world ETPs have hundreds of
  variables and hundreds of binary constraints and
  limits
• Domain sizes can be large (tens of employees)
• Experiments on random CNs and ETPs --gt only very
  easy problems are solvable in reasonable times

6
Plan of Research

• Introduction
• Timetabling Problems and Employee Timetabling
  Problems ETPs
• A general ETP model
• Constraints of ETPs the search problem
• Search algorithms for ETPs (TTPs)
• Complete search algorithms intelligent
  backtracking
• Stochastic Search
• Learning heuristics of Search
• Heuristics of search
• Cost functions and their parts
• Dynamically changing cost functions
• Automatic analysis of parameters of the
  algorithms
• High-level search strategy (heuristic) selection

7
Plan of Research (II)

• Timetabling Problems and Employee Timetabling
  Problems ETPs
• What are Timetabling problems TTPs
• Examples
• Solving methods (literature)
• Difficulty Success in solving Size
• Employee Timetabling Problems - ETPs
• Some examples
• Many families of problems
• A general model of shift/task assignment
• Constraints of ETPs the search problem

8
Plan of Research (III)

• Search algorithms for ETPs (TTPs)
• Complete search algorithms intelligent
  backtracking
• Stochastic Search local search
• Performance on ETPs
• Solving strategies from former work examples

9
Plan of Research (IV)

• Learning heuristics of Search
• Learning applied to search strategies
• A zoo of (tunable) search strategies ?
• Automatic analysis of parameters of the
  algorithms
• Cost functions and their parts
• Dynamically changing cost functions
• High-level search strategy (heuristic) selection
• Can families of similarity be found ?
• Learning the parameters of search strategies

10
Local search - Definition

• Form a search space from all possible assignment
  states
• Move on this space, guided by a cost function,
  attempting to improve the current state
• Local search algorithms move locally, in a
  limited neighbourhood (limited moves)
• Stop if a goal state has been reached or if some
  criterion on iterations/improvements holds
• For pure search problems the cost function can
  be the number of constraints violations and the
  goal has cost 0

11
Representing ETPs - for LS

• Use the known structure of the problem for a
  useful representation, for all solving methods
• Good representation of the constraints - for
  computational efficiency
• An intuitive representation of the assignment
  state - for design of a local search algorithm
• --gt an assignment table
• rows for employees
• columns for shifts
• assigned values are tasks

12
Representing Assignment States

13
Assignment Matrix - Features

• Am x n an integer valued matrix
• cells Aij are assigned values of tasks Tm
• Represents the assignment state and can be used
  for fast checking of constraints
• Employees can only be assigned once per shift
• Requirements are counted on Columns
• Unavailabilities are denoted by -1 (---)
• Workloads are simple counts on rows
• Abilities do not need a special representation
• Regulations use additional data structures, to
  point to cells that have to be counted

14
Local searching on ETPs

• Consider the set of legal states (requirements
  satisfied) as the local search space
• Cost function - number of violated constraints
• a move - Replacement
• for a given shift Sh and two employees Ei and Ej
  such that Aih k and Ajh 0, Replace(h,i,j)
  results in the same state but with Aih 0 and
  Ajh k
• Initial states can be constructed greedily by
  assigning employees to all required ltSj ,Tkgt

15
Generalized Hill climbing

• Extend search space to include all partial
  assignments by adding two moves
• Insert lth,i,kgt adds the assignment Aih k
• Delete lth,igt generates the change from Aih ?
  0 to Aih 0
• Neighbourhoods of states are generated by using
  all three moves
• All assignment states on the search tree of an
  exhaustive algorithm are now included in the
  search space

16
The Cost Function for GHC

• One component stays - number of violations
• Another obvious component is the number of
  missing assignments
• For both of these components the clear goal is
  zero cost
• Violations exist also for a complete assignment,
  missings are added cost of a partial assignment
• We add a third component that asseses the
  possibility for the partial assignment to be
  completed - the look-ahead factor

17
Cost, adaptivity, stopping

• Look-ahead sums up all conflicting assignments
  with missings - goes to zero for full assignments
• Missings and violations are the main parts of
  the cost function
• Their relative weights are adapted during
  search, based on violations counts
• Search halts when goal is reached or a number of
  iterations is passed with no improvement
• Stopping based on original cost function...

18
Search procedure

• Moves can be selected by scanning the
  neighbourhoods in several ways
• completely (best neighbour) - steepest
• randomly shalowest
• best replacement for a random (shift,employee)
• best replaced employee for a random shift
• Comparing LS algorithms, uses different moves,
  same selection, different algorithms
• To enable a fair comparison, all initial states
  are complete assignments

19
Comparative Experiments

• Two types of real world problems -
• nurse scheduling 43 shifts 29 employees 102
  needed assignments
• production-line 21 shifts 50 employees 280
  needed assignments
• Best results are obtained for GHC and have 58
  of the problems solved (38 for HC) and an
  average of 0.5 violations for all problems (0.9
  for HC)
• Best results for HC and GHC are for different
  steepness of climbing -
• best replacement for random shift for GHC
• random selection for HC

20
Discussion of experiments

• The look-ahead factor, its particular form
  (biased), and the specific mechanism of cost
  adaptivity play a role in the quality of the
  solutions
• The random selection of move types also plays a
  role, favorable for majority of replace
• The problems were not solvable with exhaustive
  search, implemented in ECLiPSe
• Tabu search performed worse than GHC
• Combinations of constructive search interleaved
  with local search performed worse than GHC
• The ECLiPSe program generated the initial
  state..

21
Conclusions (and some more details)

• A very general model for ETPs
• Standard file formats for ETPs
• take a look at the ETP homepage at BGU
• Efficient representation for assignments and
  complex constraints
• detailed in a longer paper - ETP homepage
• Constraint networks of ETPs are nonbinary, large
  --gt not solvable by exhaustive search
• Local search uses partial assignments, good
  performance, benefits from the representation

22
Employee Timetabling - CNs

• ETPs have typical complex constraints
• at most 3 Night_Shifts in 2 weeks
• no more than 3 E.R. duties in a row
• one of the doctors must be senior, if a student
  is assigned to the shift --gt n-ry constraint
• Casting ETPs in CLP enables a representation of
  limit constraints as cummulative constraints
• Problem Backtracking methods are normally
  realized with binary constraints only

23
Local Search Methods

• Local search methods are types of hill-climbing
  and are defined by
• The search space (all possible assignments)
• Neighbourhoods of assignment states
• A cost function, to guide the navigation of the
  search space -
• number of violations for a pure search problem
• additional objective function for optimization

24
Solving real-world ETPs

• Smart representation improves the (pure) search
  performance
• --gt Exhaustive search fails badly for real-world
  (pure) ETPs
• Local search methods (hill-climbing) work well
  for large-sized (pure) ETPs
• The (specific) representation of ETPs generates
  a useful neighbourhood structure for LS, with
  meaningful moves

25
Further steps for solving ETPs

• Checking complex constraints in O(1)
• Where are the hard problems ?
• Understanding the features of ETPs
• Similarities with all TTPs
• Can all complex constraints be smartly
  represented ?
• Adaptive Local Search for Solving TTPs